Subgroups — #30856072

Order 17152.

818 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	2	yes	maximal
2	#22560350	C₂	1	4	yes	maximal, complemented
3	#22560347	C₂	1	4	yes	maximal, complemented
4	#22559936	C₂	1	4	yes	maximal
5	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	4	yes	—
7	#16863420	—	2	4	no	—
8	#16863420	C₂²	1	4	yes	—
9	#16863420	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
11	#16863161	C₂²	1	16	yes	—
12	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×M₁₆×C₆₇	C₄	1	4	yes	—
15	C₄×M₁₆×C₆₇	C₄	1	4	yes	—
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×#11761612	—	2	4	no	—
19	C₂×#11761612	—	2	16	no	—
20	C₂×#11761612	C₂³	1	8	yes	—
21	C₂²×C₆₇⋊C₈	—	2	4	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
24	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₆₇	—	2	8	no	—
27	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	D₄	1	4	yes	—
32	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	#97×C₆₇	C₂×C₄	1	8	yes	—
36	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
38	#11761612	—	2	16	no	—
39	#11761612	—	4	8	no	—
40	C₂×C₆₇⋊C₈	—	2	8	no	—
41	C₂×C₆₇⋊C₈	—	4	8	no	—
42	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	—	4	4	no	—
44	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	#41	1	8	yes	—
52	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
53	M₁₆×C₆₇	—	2	16	no	—
54	C₂×C₈×C₆₇	—	2	8	no	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
56	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
57	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
58	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
61	C₄²×C₆₇	C₂×D₄	1	2	yes	—
62	C₄²×C₆₇	C₂×D₄	1	2	yes	—
63	C₂³×C₆₇	—	4	2	no	—
64	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
65	C₂³×C₆₇	#125	1	4	yes	—
66	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	—	2	8	no	—
68	C₂×C₄×C₆₇	—	2	4	no	—
69	C₂×C₄×C₆₇	—	4	4	no	—
70	C₂×C₄×C₆₇	—	4	2	no	—
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
73	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₆₇	#117	1	8	yes	—
76	C₂×C₄×C₆₇	#124	1	4	yes	—
77	C₂×C₄×C₆₇	#124	1	4	yes	—
78	C₂×C₄×C₆₇	#125	1	4	yes	—
79	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
82	C₈×C₆₇	—	2	8	no	—
83	C₆₇⋊C₈	—	4	8	no	—
84	C₂²×C₆₇	—	2	4	no	—
85	C₂²×C₆₇	—	2	2	no	—
86	C₂²×C₆₇	—	4	4	no	—
87	C₂²×C₆₇	#382	1	4	yes	—
88	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₆₇	C₂×#125	1	2	yes	—
90	C₄×C₆₇	—	2	4	no	—
91	C₄×C₆₇	—	2	2	no	—
92	C₄×C₆₇	#382	1	4	yes	—
93	C₄×C₆₇	#430	1	4	yes	—
94	C₄×C₆₇	C₂×#117	1	2	yes	—
95	C₄×C₆₇	C₂×#124	1	2	yes	—
96	#20769	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	—	2	2	no	—
98	C₂×C₆₇	—	2	4	no	—
99	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
100	C₂×C₆₇	C₂×#430	1	2	yes	—
101	C₂×#423	—	67	2	no	—
102	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
103	#1838	—	67	4	no	—
104	#1837	—	67	4	no	—
105	#1738	—	67	4	no	—
106	C₆₇	#20769	1	1	yes	characteristic, complemented, Sylow
107	C₂²×M₁₆	—	67	4	no	—
108	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
109	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
110	#423	—	67	4	no	—
111	#423	—	67	4	no	—
112	#423	—	134	4	no	—
113	C₂×C₄⋊C₈	—	134	4	no	—
114	C₄×M₁₆	Dic₆₇	1	4	yes	—
115	C₄×M₁₆	Dic₆₇	1	4	yes	—
116	C₂×#97	C₂×D₆₇	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
118	#341	—	67	16	no	—
119	C₂³×C₄	Dic₁₃₄	1	2	yes	—
120	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
121	C₂×M₁₆	—	2	8	no	—
122	C₂×M₁₆	—	67	8	no	—
123	C₂×M₁₆	—	134	16	no	—
124	C₂×M₁₆	—	134	4	no	—
125	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
126	C₂²×C₈	—	134	4	no	—
127	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
128	C₂×C₄²	Dic₁₃₄	1	2	yes	—
129	C₂×C₄²	D₂₆₈	1	4	yes	—
130	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
131	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
132	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
134	C₄⋊C₈	—	134	8	no	—
135	C₄⋊C₈	—	134	8	no	—
136	#97	C₂×Dic₆₇	1	8	yes	—
137	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
138	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
139	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
140	C₂²×C₄	—	4	4	no	—
141	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
142	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
143	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
144	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
145	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
146	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
147	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
148	C₂²×C₄	#11761628	1	8	yes	—
149	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
150	M₁₆	—	2	16	no	—
151	M₁₆	—	268	8	no	—
152	M₁₆	—	134	16	no	—
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	—	268	8	no	—
155	C₂×C₈	—	134	8	no	—
156	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
157	C₄²	C₄⋊Dic₆₇	1	4	yes	—
158	C₄²	C₄⋊Dic₆₇	1	4	yes	—
159	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
160	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
161	C₄²	C₂²×Dic₆₇	1	2	yes	—
162	C₄²	C₂×D₂₆₈	1	2	yes	—
163	C₄²	C₂×D₂₆₈	1	2	yes	—
164	C₂³	—	4	2	no	—
165	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
166	C₂³	#14524035	1	4	yes	—
167	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
173	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
176	C₂×C₄	#14524002	1	8	yes	—
177	C₂×C₄	#14524031	1	4	yes	—
178	C₂×C₄	#14524031	1	4	yes	—
179	C₂×C₄	#14524035	1	4	yes	—
180	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
181	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
182	C₂×C₄	C₂×#11761628	1	4	yes	—
183	C₈	—	2	8	no	—
184	C₈	—	268	8	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	4	4	no	—
188	C₂²	#16863262	1	4	yes	—
189	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
190	C₂²	C₂×#14524035	1	2	yes	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	#16863262	1	4	yes	—
194	C₄	#16863445	1	4	yes	—
195	C₄	C₂×#14524002	1	2	yes	—
196	C₄	C₂×#14524031	1	2	yes	—
197	C₂	—	2	4	no	—
198	C₂	—	2	2	no	—
199	C₂	C₂×#16863262	1	1	yes	characteristic
200	C₂	C₂×#16863445	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#20769
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #22560350	C₂	semidirect
3. #22560347	C₂	semidirect
106. C₆₇	#20769	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16863420	2
#22560350	2
#22560347	2
#22559936	2
C₂×C₄×M₁₆×C₆₇	2
#20769	67